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Chapter 6: Problem 30

Find a formula for \(\sin \left(\theta+\frac{\pi}{2}\right)\).

Short Answer

Expert verified
The formula for \(\sin\left(\theta+\frac{\pi}{2}\right)\) is \(\cos\theta\).

Step by step solution

01

State the sine and cosine values for the given angles

Before applying the sine addition formula, let's find out the sine and cosine values for the angles \(\theta\) and \(\frac{\pi}{2}\): \(\sin\theta=\sin\theta\) (since \(\theta\) is a general angle) \(\cos\theta=\cos\theta\) (since \(\theta\) is a general angle) For \(\frac{\pi}{2}\), we have: \(\sin\frac{\pi}{2}=1\) (as the sine of 90 degrees is 1) \(\cos\frac{\pi}{2}=0\) (as the cosine of 90 degrees is 0)
02

Apply the sine addition formula

Now, we apply the sine addition formula as follows: \(\sin(\alpha+\beta) = \sin\alpha\cos\beta+\cos\alpha\sin\beta\) Replacing \(\alpha\) with \(\theta\) and \(\beta\) with \(\frac{\pi}{2}\), we get: \[\sin\left(\theta+\frac{\pi}{2}\right) = \sin\theta\cos\frac{\pi}{2}+\cos\theta\sin\frac{\pi}{2}\]
03

Substitute the sine and cosine values

Now, let's substitute the sine and cosine values that we found in Step 1: \[\sin\left(\theta+\frac{\pi}{2}\right) = (\sin\theta)(0)+(\cos\theta)(1)\]
04

Simplify the expression

Now, we can simplify the expression by removing unnecessary terms: \[\sin\left(\theta+\frac{\pi}{2}\right) = \cos\theta\] Thus, the formula for \(\sin\left(\theta+\frac{\pi}{2}\right)\) is \(\boxed{\cos\theta}\).

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